Automultiscopic displays based on orbitalangular momentum of light

Xuefeng Li1, Jiaqi Chu1, Quinn Smithwick2 and Daping Chu1

1 Photonics and Sensors Group, Department of Engineering, University of Cambridge, UK2Disney Research, Glendale, CA, USA

E-mail: dpc31@cam.ac.uk

Received 28 January 2016, revised 14 March 2016Accepted for publication 5 April 2016Published 20 July 2016

AbstractOrbital angular momentum (OAM) of light has drawn increasing attention due to its intriguinglyrich physics and potential for a variety of applications. Having an unbounded set of orthogonalstates, OAM has been used to enhance the channel capacity of data transmission. We proposeand demonstrate the viability of using OAM to create an automultiscopic 3D display. Multi-viewimage information is encoded using an OAM beam array, then sorted into different viewdirections using coordinate transformation elements. A three-view demonstration was achievedto encode and decode 9×9 pixel images. These demonstrations suggest that OAM couldpotentially serve as an additional platform for future 3D display systems.

Keywords: orbital angular momentum, display, automultiscopic 3D

(Some figures may appear in colour only in the online journal)

Introduction

The recent convergence of advances in display technology,image capturing, signal processing and optical communica-tions has generated a paradigm shift in how information iscollected, conveyed and rendered. This area of research hasbeen developed rapidly due to the widespread use of flat paneldisplays [1] and spatial light modulators (SLMs), especiallythose based on liquid crystals (LC) [2]. In particular, theproduction of and consumer interest in 3D content andtechnologies, such as virtual reality and augmented reality,have been increasing exponentially in the past few years. 3Ddisplay technology combines both rendering and conveyanceof spatial information [3–5]. The state-of-the-art autostereo-scopic 3D display technologies can be classified into threebroad categories: (1) multiview 3D display [6], (2) volumetric3D display [7], and (3) digital holographic display [8].Among existing multiview encoded 3D display approaches,there are various mechanisms including [4]: occlusion-basedtechnologies (parallax barrier, time-sequential aperture,moving slit, and cylindrical parallax barrier), refraction-based(lenticular sheet, multiprojector, and integral imaging),reflection-based, diffraction-based, and eye tracking. One ofthe most significant drawbacks in above methods is the trade-off between image resolution in each view and the number of

viewpoints [9]. Although using a display panel with smallerpixel size (∼5 μm) could potentially increase the viewingangle while keeping the display resolution, the achievablenumber of views is limited thus compromising smooth par-allax viewing. In addition to multiview approaches, a numberof binocular (two-view) stereoscopic 3D display techniqueshave already been commercialized due to their high qualityand low cost. Nevertheless, the need to wear polarization orshutter glasses is a major concern; polarization supports onlytwo-views thus supporting only a single correct perspectiveand requiring tracking for varying viewpoints; while shutterglasses require active glasses.

Nevertheless, advancements in fundamental science maypave the way for 3D displays with totally different technol-ogies. As we know, the wavefront completely characterizesthe radiance flowing through all the points in a plane, in allpossible directions, and of all possible curvatures. The ulti-mate goal of 3D display systems is to faithfully reproduce thewavefront, but it is almost impossible to practically realizedue to its huge information requirements. Instead, a sub-sampling of views is implemented to mimic the 3D infor-mation. Different views are encoded and multiplexed invarious means, then separated when the information isreceived by the observer. The use of multiplexing anddemultiplexing in free space and optical fiber

Journal of Optics

J. Opt. 18 (2016) 085608 (8pp) doi:10.1088/2040-8978/18/8/085608

2040-8978/16/085608+08$33.00 © 2016 IOP Publishing Ltd Printed in the UK1

communications has benefited from the encoding/decodingusing light’s various degrees of freedom [10, 11]. Amongthem include wavelength/frequency, amplitude/phase, timesequence, and polarization of light, each of which can besolely or jointly implemented to achieve high data-carryingcapacity. Similarly, the development of multiview 3D displaytechniques has been focused on separating view channelswith respect to independent variables, e.g. time multiplexing(time) [12] and integral imaging (space) [13].

Angular momentum is one of the most fundamentalphysical quantities in both classical and quantum physics[14]. It has long been known that the circular polarizationstate of electromagnetic waves (e.g. visible light) is associatedwith spin angular momentum [15]. In contrast, the orbitalcomponent of angular momentum in optics is affiliated with avortex-like phase profile qe li [16], where l is the topologicalcharge, and q is the azimuthal angle. Optical beams with thishelical phase have been shown to possess a well-definedorbital angular momentum (OAM) of l per photon. In recentdecades, OAM of light has drawn tremendous attention [17]due to its potential application in communication [18, 19],imaging [20], optical tweezers [21], etc. In particular, OAM isvery attractive for data transmission due to its nature ofunlimited number of potential states so that OAM beams cancarry an infinite amount of data in principle. It has beendemonstrated that free-space and optical fiber communicationwith OAM multiplexing/demultiplexing can work at broad-band frequency with much higher data rate capacity [18, 19].There are two approaches to increase the channel capacitywith OAM, either by encoding information as OAM states ofthe beam or by using OAM beams as information carriers formultiplexing. In the field of 3D display, if we can encode eachview with one of the OAM eigen-state, a 3D display with

unlimited views can theoretically be constructed, as illustratedin figure 1.

In this paper, we present an autostereoscopic multi-view3D display architecture based on the multiplexing anddemultiplexing of OAM-carrying beams. First, we describethe concept of encoding of 2D image information using OAMmodes, multiplexing of different OAM channels, thensimultaneous separation of different images into variousviewing angles. Experimentally, a three-view proof-of-con-cept display is demonstrated, where three 9×9 pixel imagesare encoded and spatial separated using OAM beam arrays.Our approach efficiently breaks the dependence betweennumber of available views and the image resolution. More-over, the compatibility between OAM based 3D display anddata communication may provide the possibility for all-opti-cal collection, transmission, and rendering of 3D information.This might pave the way for the next generation of multiview3D display technologies.

Concept and optics design

For data transmission both in free space or optical fibercommunications, OAM modes serve as a carrier of informa-tion and they are regarded as an additional degree of freedomwhich is complementary to the existing multiplexing tech-nologies, permitting a greatly increased data-carrying capacity[19]. However, the information is generally encoded timesequentially as a bit-stream for each OAM channel. Thus a2D static image is typically transmitted pixel by pixel [22] ina raster order, similar to the reciprocal way of a CRT display.Generally, a 2D image can be expressed as a two-dimensionalfunction associated with a complex field amplitude. In con-ventional imaging system, this 2D information is imprinted

Figure 1. Schematic illustration of a multiple-view 3D display using orbital angular momentum of light.

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onto the optical beam using an SLM. Due to the lack ofcomplex phase structures in zero-order Hermite–Gaussianbeams, the encoded images can be relayed without muchdistortion. However, direct superposition of the complex fieldof an image onto an OAM helical phase structure would leadto the breakdown of OAM eigen-modes in the far field andtherefore large crosstalk between different channels. Never-theless, there are another two mechanisms to incorporate the2D information into OAM modes without disrupting theorthogonality between distinct states. (1) Convolutionbetween an image and an OAM mode, which has beeninvestigated extensively in spiral phase contrast imaging/microscopy [23]; (2) encoding the 2D amplitude distributionas an array of OAM beams, where each OAM element cor-responds to one pixel of the encoded image. In this paper weemploy the array approach due to the fact that the convolutionmethod requires complex optics in order to separate differentviews, while the latter one can benefit from the OAM modesorting using coordinate transformation components. Once anumber of images are encoded with different OAM modes,they can be multiplexed into one optical beam bundle usingbeam splitters, as shown in figure 2(a). Unlike their use incommunication systems, multiplexed OAM beams wouldpropagate no more than 5 m before the information is decodedby an OAM sorter and projected to the imaging plane. Thusthe effect of air turbulence can be neglected in this case [24].

To achieve the separation between different views, thecoordinate transformation, which maps the azimuthal comp-onent p-0 2( ) into the Cartesian coordinates, can effectivelyunwrap a donut-like OAM beam into a rectangular shape witha tilted wavefront, as proposed and demonstrated by Padgett’sgroup [25]. The OAM sorter performs a mapping from theinput plane (x, y) to the output plane (u, v), which can bedescribed as: =v a y xarctan ( ) and

= - +u a x y bln .2 2( ) And the phase profile of thetransforming optical element is expressed as:

⎜ ⎟⎡⎣⎢⎢

⎛⎝

⎞⎠

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤⎦⎥⎥

fpl

=

´ -+

+

x ya

f

yy

xx

x y

bx

,2

arctan ln 1

1

2 2

( )

( )

and the phase aberrations caused by the transformation can becorrected by a second element as:

⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠f

pl

= - -u va

f

u

a

v

a,

2exp cos , 22 ( ) ( )

where a=d/2π scales the size of the transformed beam andb determines the position of the output beam. λ is thewavelength of the light, and f is the focal length of the lensbetween the above two elements. By constructing an array ofsorter and corrector pairs, we can effectively route differentOAM beam arrays into different directions and generate amultiview 3D image.

The designed phase profile can be generated usingstandard phase-only SLMs, but diffraction based SLMs havelimited diffraction efficiency and the pixelated nature of SLM(pitch∼10 μm) would result in poor performance of OAMsorting [25]. Instead we use refractive elements to carry outthe sorting process [26] and the surface profile of the sortercan be arbitrarily manufactured using diamond machining. Inaddition, the Fourier transform lens can be incorporated intothe first element and corrected on the second element byanother lens term. By carefully tailoring the parameters, thedesigned freeform surfaces can be fabricated on the front andback sides of a single optical plate respectively, which allowsfor compactness and precise alignment of optics, as shown infigure 2(b).

Simulation

First, we consider the generation of OAM beam arrays. Avariety of ways have been implemented to generate singleOAM beam with helical phase profile, such as spiral phaseplate (SPP) [27], q-plate [28], combination of Hermite–Gaussian modes [16], and SLMs [29]. A computer-generatedhologram can be used to transform the pseudo plane-wavelaser beam into one with exotic phase structure. In this paper,we use the SPP to generate Laguerre–Gaussian (LG) beams asOAM-carrying states with different indices l. And ABCDmatrix method [30] is applied to simulate the propagation andwavefront shaping of optical waves. Figure 3(a) displays theintensity of the simulated LG beam when a Gaussian beamwith the waist size =w 1.0 mm0( ) passes through a SPP with

Figure 2. (a) Conceptual illustration of multiplexing images with three OAM states. (b) Designed sorter and corrector array for sorting ofOAM beam lattice. The height profiles of the sorter and corrector are shown in the insets, respectively.

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a charge =l 6 .( ) Inspired by Mirhosseini et al [31], weinvestigate the transformation of a single OAM beam to asquare lattice of optical vortices using the beam copyingdevice. As shown in figure 3(b), we use a phase-only holo-gram to make multiple coherent copies (9×9) of the LGbeams with identical amplitude and phase profiles. The phasestructure of this element can be written as:

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

åå

åå

g p l a

g p l a

g p l a

g p l a

Y =+

+

++

+

- =-

=-

- =-

=-

s mx

s mx

s ny

s ny

tansin 2

cos 2

tansin 2

cos 2, 3

m M

Mm m

m M

Mm m

n N

Nn n

n N

Nn n

1

1

[( ) ]

[( ) ]

[( ) ]

[( ) ]( )

where +M2 1, and +N2 1 are the number of copies of theOAM beam in horizontal and vertical directions, respectively.s is the angular separation between adjacent copy, while gm n,and am n, are relative amplitude and phase parameterscorresponding to separate diffraction orders. These para-meters are optimized to achieve uniformity between themultiple copies. Then light propagating through the diffrac-tion beam splitter and a corrector is collected by an alignedlenslet array that directs the OAM beam array onto an imagemask. The mask is also aligned with the beam array andproduces the desired OAM-encoded view image. Figure 3(c)

shows the transmitted image of the letter ‘P’ consisting ofOAM pixels.

The sorter converts the azimuthal phase gradient intotilted plane waves (i.e. linear phase gradient) with the tilt of aresulting plane wave proportional to the pitch of a spiralwavefront e .lxi The phase correction plate removes residualphase aberrations caused by the sorter element. Additionalfan-out and phase correctors can be used to reduce thecrosstalk between OAM modes. Figure 3(d) shows theintensity of a coordinate transformed OAM beam =l 6( ) justafter the phase corrector. Note, the transformed optical beamis not a perfect rectangular shape, but has a sinusoidal dist-ortion due to the nonzero skew angle of OAM beams [25].The parameters of the sorter are: =d 8 mm, and =b 0.004.Passing the tilted wavefronts through a lens, the lateralposition (tl) of each focused beam varies with the tilt of theincoming plane wave. Hence the propagation direction ofdifferent transformed OAM beams can be expressed as:

q = = ltan ,t

f

l

dl where θ is the divergence angle. Figure 3(e)

shows the modeled intensity profile of the sorting of sixequally superposed OAM modes on the image plane, whilethe index separation between adjacent modes is D =l 4.Although, these modes are well separated with each other onthe imaging plane, the angular separation ( qD = 0.02 for=d 8 mm) is still too small compared with a conventional

multiview 3D display ( qD > 0.7 .) Therefore, both the size

Figure 3. (a) Simulated intensity profile of a LG beam with index =l 6. (b) The modeled intensity of a 9×9 array of OAM-carrying LGbeams generated by beam copying device. (c) Transmitted image of the letter ‘P’ after passing through an image mask. (d) Intensity profile ofthe coordinate transformed OAM beam ( =l 6) directly after the corrector. (e) and (f) Simulated OAM spectra of (e) a single and (f) an arrayof superposed LG modes with = - - -l 10, 6, 2, 2, 6, 10. The scale of the white bar is 1.0 mm.

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of OAM beam and sorter parameter should be reduced toenable a noticeable viewing angle separation. Figure 3(f)shows the intensity profile of the sorted OAM array con-stituted of equally weighted LG beams with the scaling factor=d 1 mm on each sorter pixel. If each OAM array is coded

with a different image mask, we are able to construct a sixview 3D display using the sorter array.

Experiment

In the experiment, we generated the input OAM test modes byuse of a SPP, which was manufactured using laser directwriting method by RPC Photonics. The SPP has equallyspaced sub-areas ( ´10 mm 10 mm) with an adjustable heli-cal index l ranging from −8 to 8. The Gaussian beam emittedfrom a 632.8 nm HeNe laser was expanded by a pair of lenses( =f 10 mm,1 =f 100 mm2 ) and illuminated onto the SPP,producing the desired OAM state sequentially. Two adjus-table apertures were placed in front and at the back of the SPPrespectively to remove the noise outside the area of theexpanded Gaussian and LG beam. Figure 4(a) shows ameasured far-field intensity pattern of an OAM-carrying beamwith a charge =l 6. In order to produce an OAM beam arraywith equally distributed energy and high diffraction effi-ciency, we encoded a Dammann grating structure [32] onto a

LC on silicon SLM (256 gray level, 15 μm pixel pitch). Theadvantage of using the Dammann grating instead of fan-outcomponent is that the beam copying process discussed aboverequires both transformation and correction [31], which addscomplexity into the optics alignment. In addition, the phasestructure of a Dammann grating is binary, allowing for costeffective fabrication using standard photo-lithography andeasy integration to other systems. The phase values on aDammann grating, either 0 or π, are encoded on a series oftransition points within each period. The positions of thesetransition points are optimized to achieve maximum diffrac-tion efficiency. We designed the Dammann grating (period:150 μm) according to the transition points listed in [33]achieving a 9×9 array of OAM carrying beams, as shown infigure 4(b). The observed OAM array exhibits good qualityperiod and intensity distributions. The Airy ring pattern sur-rounding each OAM pixel was due to the use of circularapertures in our set up.

The OAM sorter and corrector were designed to increasethe measurement bandwidth by use of a large aperture

=d 18 mm( ) and small element distance =D 300 mm .1,2( )The two-part OAM sorting components were fabricated usinga diamond machining method by Precision Optics Laboratoryin Durham University. A Nanotech three-axis ultra-precisionlathe was used in combination with a Nanotech NFTS6000fast tool servo system to provide a fast axis superimposed on

Figure 4. (a) Measured intensity profile of a LG beam with an index =l 6. (b) The observed intensity of a 9×9 array of OAM-carrying LGbeams generated by Dammann grating. (c) Measured intensity profile of the coordinate transformed OAM beam directly after the corrector.(d)–(f) Experimentally measured OAM spectra of an array of different LG modes with (d) =l 4, (e) =l 0, and (f) = -l 4. The scale of thewhite bar is 2.0 mm.

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one of the axes of the lathe. The detailed fabrication processcan be found at [34]. The generated OAM beam propagatedthrough the elements, unwrapping the donut-like beam, andmapping the azimuthal coordinate into a transverse axis.Figure 4(c) shows the intensity of a coordinate transformedOAM beam with the charge =l 6, exhibiting good rectan-gular shape.

In order to demonstrate the sorting performance of auniformly distributed OAM lattice, we illuminated the sortedsingle OAM beam onto the Dammann grating hologramdisplayed on the SLM and a CCD camera was placed at thefocal plane of the final focusing lens. The measured OAMspectra are shown in figures 4(d)–(f), which corresponds toOAM modes = -l 4, 0, 4, respectively. The horizontal andvertical lines in these images mark the center of the zeroorder. The distortion to the desired elongated spot array arisesfrom the misalignment of the optics. A shift in the transversedirection can be observed with a distance of 65.6 μm separ-ating the adjacent OAM modes. This measured shift is con-sistent with the theoretical model = lt f ,l

dl which gives alateral displacement of 17.6 μm for an increase of OAM modedifference D =l 1. We observed a shift of pattern in thevertical direction due to the size of the OAM beam expandingwith the increase of mode number. For example, the intensityof a normal zero order Hermite–Gaussian beam has itsmaximum at the beam center, while the intensity maximum ofan LG beam lies on the ring. Therefore, we can observe ajump of spots centers in the original radial direction, as shownin figures 4(d)–(f).

After demonstrating the sorting performance of theOAM beam array, the designed image masks were insertedinto the optics for 3D display purposes. As shown infigures 5(a)–(c), a group of three image masks weredesigned with each mask designated one of the letters ‘P’,‘S’, and ‘G’. These masks were fabricated on transparentfilms with the dark areas designed to block the incident light.Figures 5(d)–(f) show the observed image on the CCD wheneach letter is encoded with an OAM state ( = -l 7, 0, 7)respectively. The shapes of the original images were wellreconstructed in each case and they are consisted of elon-gated pixels. The noise in these images was due to thescattering on the surface of the inkjet printed masks. Mostimportantly, these three images were spatially separated dueto the OAM mode difference. The transverse displacementin the horizontal axis is over 100 μm, which corresponds to aseparation angle 0.015°.

Discussion

We have demonstrated a proof-of-concept three-view 3Ddisplay based on the mode orthogonality between OAM-carrying optical beams. The difference in OAM mode indexprovides a convenient way to sort multiple images by con-structing a multiplexed OAM beam array. However, the anglebetween each views is very small 0.015( ) compared to that ofa conventional multiview 3D display > 0.7( ) due to the large

size of the sorter element parameter ( =d 18 mm .) As dis-cussed above, the separation angle can be written asq = larctan l

dand larger viewing angle can be achieved using

sorter arrays with smaller scaling parameter d. Manufactur-ing of sorting elements with size m<d 500 m is totallywithin the fabrication capabilities of our diamond machiningequipment. Sub-micron resolution of this fast tool servosystem can be obtained, which is sufficient for our opticsdesign. Also, the working distance of this 3D display can beadjusted using relaying optics such as micro-lens array.Furthermore, the effective resolution of this 3D displaysystem can be greatly improved with smaller pixel (OAMbeam and sorter) size.

In this paper we study the angular separation betweenviews, which is the basic requirement of a multiview 3Ddisplay. In addition, the coordinate transformation can beengineered to provide depth information of 3D scenes. Theconformal mapping described in this study explores the polarto Cartesian transformation, which unwraps the phase gra-dient in the azimuthal direction into a transverse one. Areciprocal process can be implemented in order to transformthe phase gradient from transverse direction to the radialdirection (Cartesian to polar) [35]. If a lens is placed behindthe transformed pattern, beams with different OAM index lcan be mapped to various depths along the beam axis. Bycombining both the angular tiling and depth mapping, a morecomprehensive 3D display system can be constructed usingimages coded with OAM states.

Moreover, we have noticed that the radial component isalso affected in the sorting process. This is because the beamwaist enlarges with the increase of the OAM angular index l.If we are able to separately resolve the radial and angularcomponents of an OAM-carrying beam, parallax viewing inboth the horizontal and vertical directions is possible. Toachieve this, Fourier transform of Bessel beams are choseninstead of SPP generated LG beams, which can be producedby programming the phase pattern on phase-only SLM [36].The annular ring structure containing an azimuthal order canbe arbitrarily engineered for simultaneous separation of OAMradial and azimuthal index by use of the sorter elementsdescribe above.

In addition, the size of these pixels has no lower limit inprinciple. A large OAM capacity can be maintained byreducing the beam size (pixel size) and the element separationsimultaneously. In practice, the pixel size here is usuallydetermined by the resolution of the manufacturing processand the size of the display pixel can be manufactured in thetens of micrometers.

Finally, only the stereoscopic information is encoded anddecoded here. However, the Cartesian-to-Polar coordinatetransformation can be added to include the depth informationby using two more elements. The view angle separationbetween adjacent modes is dependent on the element size andit increases with the reduction of the pixel size. Another wayto increase the viewing angle separation is to reduce thenumber of OAM modes used in the 3D display.

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Conclusion

In conclusion, we have proposed and demonstrated, boththeoretically and experimentally, a new multiview 3D displayarchitecture using OAM based multiplexing and sorting.Information of different views are encoded into the OAMcarrying beam array and decoded with a sorting elementarray. The phase gradient provided by OAM provides agenuine tool for sorting different modes and allows for mul-tiview display. A three-view experiment was performed todemonstrate the pixel based 3D display, and sufficient toshow capabilities beyond only two-views using polarizationmultiplexing. The separation angle between views can befurther increased by using smaller sorter components. Thisdesign of autostereoscopic display may pave the way forfuture 3D display technologies.

Acknowledgments

This research was performed under a joint collaborationbetween Disney Research and the University of Cambridgethrough the CAPE consortium. XL, JC and DC would like tothank the UK Engineering and Physical Sciences ResearchCouncil (EPSRC) for the support through the Platform Grantfor Liquid Crystal Photonics (EP/F00897X/1).

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Figure 5. (a)–(c) Encoded image masks of letter ‘P’, ‘S’, and ‘G’, which are imprinted on transparent films. Each letter is associated with adistinct OAM mode with ‘P’ corresponds to OAM state =l 7, ‘S’ to =l 0, and ‘G’ to = -l 7, respectively. (d)–(f) Observed image patternson the CCD camera. The position of the measured image shifts on the horizontal axis, depending on the value of OAM state on which theimage is encoded.

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